Interior Points
By Sarah Richards
I'm learning analysis using the Rudin's book, and sometimes the definitions make me wonder and leave me quite puzzled...
So, interior points: a set is open if all the points in the set are interior points. However, if a set has a point inside it, surely it will always have a neighborhood (or a small ball) that will be contained in the set. So, what keeps all the points from being interior points? (points inside the set I mean)
Also, second question: is a limit point an interior point?
Thank you!
$\endgroup$3 Answers
$\begingroup$To answer your other question: a limit point of a set $A$ can be an interior point of $A$, but it need not be. Let $A=[0,1)$ with the usual topology, for instance. Then $\frac12$ is a limit point of $A$ that is also an interior point of $A$, and $0$ and $1$ are limit points of $A$ that are not interior points of $A$. For another example, in the real line with the usual topology every point is a limit point of $\Bbb Q$, and no point is an interior point of $\Bbb Q$.
$\endgroup$ 3 $\begingroup$However, if a set has a point inside it, surely it will always have a neighborhood (or a small ball) that will be contained in the set.
Not true: consider $\Bbb R$ with Eucledian topology and a set $A = \{0\}$. No balls of positive radius around $0$ are contained in $A$.
$\endgroup$ 0 $\begingroup$$A$ be a closed set and $D = \overline{A^\circ}$. Prove that $D^\circ=A^\circ$.
Notation:
$A^\circ$: interior of $A$. $\overline{A}: $ closure of $A$.
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